A Comparison of Rates of Convergence of Two Inexact Proximal Point Algorithms

We compare the linear rate of convergence estimates for two inexact proximal point methods. The first one is the classical inexact scheme introduced by Rockafellar, for which we obtain a slightly better estimate than the one given in [16]. The second one is the hybrid inexact proximal point approach introduced in [25, 22]. The advantage of the hybrid methods is that they use more constructive and less restrictive tolerance criteria in inexact solution of subproblems, while preserving all the favorable properties of the classical method, including global convergence and local linear rate of convergence under standard assumptions. In this paper, we obtain a linear convergence estimate for the hybrid algorithm [22], which is better than the one for the classical method [16], even if our improved estimate is used for the latter.